Proposal of time domain impedance spectroscopy to determine precise dimensionless figure of merit for thermoelectric modules within minutes

Several techniques exist that use a thermoelectric element (TE) or module (TM) to measure precise dimensionless figure of merit (zT), both qualitatively and quantitatively. The techniques can be applied using both alternating (AC) and direct current (DC). Herein, the transient Harman (TH) and impedance spectroscopy (IS) methods were investigated as direct zT measurement techniques using identical TM, which showed that zT at 300 K was 0.767 and 0.811 within several minutes and several hours, respectively. The zT values differed despite the use of the same TM, which revealed that measuring ohmic resistance using DC and pulse DC is potentially misleading owing to the influence of Peltier heat on current flow. In this study, time domain impedance spectroscopy (TDIS) was proposed as a new technique to measure zT using proper DC and AC. zT obtained using TDIS was 0.811 within several minutes using the time and frequency domains, and was perfectly consistent with the result of the IS method. In conclusion, the TDIS is highly appropriate in estimating zT directly using only proper electrometric measurements, and without any heat measurements.

Proposal of time domain impedance spectroscopy to determine precise dimensionless figure of merit for thermoelectric modules within minutes

Yasuhiro Hasegawa * & Mai Takeuchi
Several techniques exist that use a thermoelectric element (TE) or module (TM) to measure precise dimensionless figure of merit (zT), both qualitatively and quantitatively. The techniques can be applied using both alternating (AC) and direct current (DC). Herein, the transient Harman (TH) and impedance spectroscopy (IS) methods were investigated as direct zT measurement techniques using identical TM, which showed that zT at 300 K was 0.767 and 0.811 within several minutes and several hours, respectively. The zT values differed despite the use of the same TM, which revealed that measuring ohmic resistance using DC and pulse DC is potentially misleading owing to the influence of Peltier heat on current flow. In this study, time domain impedance spectroscopy (TDIS) was proposed as a new technique to measure zT using proper DC and AC. zT obtained using TDIS was 0.811 within several minutes using the time and frequency domains, and was perfectly consistent with the result of the IS method. In conclusion, the TDIS is highly appropriate in estimating zT directly using only proper electrometric measurements, and without any heat measurements. www.nature.com/scientificreports/ ω TE Characteristic frequency of thermoelectric or thermo-module using impedance spectroscopy (IS) (rad/s) ω R2C Characteristic frequency of thermo-module using R2C approximation (rad/s) Z(ω) Impedance as a function of angular frequency at angular frequency ω (Ω) Z R2C (ω) Impedance as a function of angular frequency at angular frequency ω applying R2C approximation (Ω) Z mea (ω) Measured impedance as a function of angular frequency at angular frequency ω (Ω) R ohm Ohmic resistance (Ω) R AC AC resistance with an alternating current by Harman method (Ω) R DC DC resistance with a direct current by Harman method (Ω) R TE Thermoelectric resistance (Ω) R ohm + R TE Sum of ohmic (R ohm ) and thermoelectric (R TE ) resistances (Ω) C TE Thermoelectric capacity in system (F) t

List of symbols
Time ( www.nature.com/scientificreports/ Thermoelectric materials and elements (TEs) that can convert a temperature gradient into electricity (Seebeck effect) or electricity into a temperature gradient (Peltier effect) have drawn significant attention as a key technology for renewable energy 1,2 . The performance and energy efficiency of TEs at the temperature T have been described as a function of dimensionless figure of merit zT, where z (= S 2 /ρκ) is the function of Seebeck coefficient (S), resistivity (ρ), and thermal conductivity (κ) 1-3 . The zT values are typically estimated using two different TEs such as a rectangular solid for the measurement of S and ρ, and a thin disk for the measurement of κ 3 ; however, the ideal estimation of the zT requires the use of the same TE or identical material. A direct zT measurement technique using a rectangular solid of the TE was proposed by Harman et al. in 1958. The Harman method 4-6 uses AC resistance R AC with an alternating current (AC) and DC resistance R DC with a direct current (DC), based on which zT is expressed as zT = R DC /R AC − 1. However, the applicability of the method is limited owing to lack of information on the frequency of the AC and suitable magnitude of the AC and DC into the TEs. Another approach to directly estimate zT using the same TE is a technique called impedance spectroscopy (IS), which uses the frequency domain based on an one-dimensional heat conduction equation [7][8][9][10][11][12][13][14][15][16] . Figure 1a,b show the schematic of Nyquist plot and the relation between its angular frequency (ω) and impedance Z(ω), based on which the zT is expressed as 10,16 where R ohm and R TE are ohmic and thermoelectric resistance, Q P ( =|S|TI) and Q J (= R ohm I 2 ) are Peltier and Joule heat, A and L are cross-sectional area and length of the TE, respectively, and η is a proportional factor associated with the heat flow of Q J to one side of the TE. In addition, zT (= R TE /R ohm ) is denoted by the ratio of certain physical parameters such as resistance because zT is dimensionless 12,16 . Therefore, the physical meaning of the zT implies a ratio of the ohmic resistance (R ohm ) to increasing resistance (R TE ) generated by the temperature difference (ΔT) between the edges of the TE, which, in turn, is caused by the Peltier heat induced by DC. In particular, measuring the resistance, which is a macro physical quantity, is easy using recent electrometric instruments with a combination of a voltmeter for DC measurement and a lock-in amplifier for AC measurement by a precision Figure 1. Schematics of (a) a Nyquist plot and (b) a frequency dependence using the IS method with Z(ω) and Z R2C (ω) 10,12,16 , and (c) time dependence of measured resistance R(t) using the TH method. An inset of (c) shows a schematic for the setting view of the TM prepared. (d) An equivalent circuit for thermoelectric element and thermoelectric module 16  www.nature.com/scientificreports/ current source. Equation (1) also shows that the condition Q P > > Q J must be met to obtain the precise value of zT. Therefore, the optimum current I opt should be I opt < <|S|T/R ohm . Moreover, the IS method is a suitable technique to determine the zT for both TEs and thermoelectric modules (TMs), which are an assembly of thermoelectric elements 10,11,16 . However, measuring the Z(ω → 0) takes several hours because a suitable characteristic frequency (ω TE ) is required at ω → 0 (or ω < < ω TE ). ω TE is denoted as a function of thermal diffusivity (α) and L of the TE, ω TE ∝ α/L 2 , and typically approaches 1 rad/s owing to the small value of κ (∝ α) of the TEs. Therefore, the angular frequency satisfying Z(ω → 0) would approximately be of the order of 10 -2 -10 -4 rad/s, depending on L 12,15 . The theory and model of the IS method clearly demonstrates that the Harman method is one of the results obtained 16 . Furthermore, the R2C approximation was applied to roughly explain the angular frequency dependence Z R2C (ω), as shown in Fig. 1a,b, using optimum current I opt , which is expressed as 12,16 The transient Harman (TH) method, which is an alternative technique derived from the Harman method, is based on the transient response of resistance R(t) of the TEs and TMs using time domain, as shown in Fig. 1c. The TH method is relatively simpler to apply in the determination of zT using results of the R(t); several researchers have also reported its applicability [17][18][19][20][21][22] . In this method, R ohm and R ohm + R TE correspond to R(t = 0) and R(t → ∞), respectively, and zT is expressed as zT = R(t → ∞)/R(t = 0) − 1 using Eq. (1). The IS and TH methods (or R2C approximation) show that an equivalent circuit of the TE and TM can be expressed using three components, namely, R ohm , R TE , and C TE (called thermoelectric capacity), as shown in Fig. 1d. C TE is related to heat capacity of not only the TE(s) but also other components constituting the TE(s) like electrodes, especially for the TM 10,12,16 .
In this study, we comprehensively investigate several techniques of direct zT estimation based on the IS and TH methods. We employ a commercial base Π-shaped TM composed of bismuth-telluride (BiTe) (inset in Fig. 1c) to avoid influences of heat leakages through lead-wires attached for measurement and their contact resistance 23 . Furthermore, we highlight the disadvantages of both methods and suggest new techniques to overcome the drawbacks of the conventional methods. Finally, we propose a suitable technique to determine the value of zT precisely and directly within several minutes using a combination of AC and DC electrometric instruments, called time domain impedance spectroscopy (TDIS), and discuss the important factors required to obtain the value of zT precisely through measurements. Figure 2 shows the frequency dependence of the measured impedance Z mea (ω) in the case of the IS method for a TM prepared at 300 K with various AC from I = 100 μA rms to 100 mA rms 16 , and the inset shows its Nyquist plot using 1 mA rms . The characteristic angular frequency (ω R2C ) using the R2C approximation given by Eq. (2) was 0.255 rad/s (= 40.6 mHz) owing to L = 1.4 mm (ω R2C ∝ 1/L 2 ). (R ohm ) IS = Z mea (ω → ∞) = Re[Z mea (f = 513 Hz)] at ω/ω R2C ~ 10 4 (or phase angle |ϕ|< 0.1°) was 480.0 mΩ, and (R ohm + R TE ) IS = Z mea (ω → 0) = Re[Z mea (f = 0.5 mHz)] at ω/ω R2C ~ 10 -2 (or |ϕ|< 0.1°) was 869.5 mΩ at a current less than 10 mA rms , satisfying Q P > > Q J . At 100 mA rms , the contribution of Q J affected Re[Z mea (ω)] and − Im[Z mea (ω)] in the lower frequency region. Therefore, Re[Z mea (ω)] (∝ I) increased marginally at 10 -3 Hz owing to Q P (= 6.9 mW) ~ Q J (= 4.8 mW) using the representative magnitude of S (= − 231 μV/K at 300 K) corresponding to BiTe standard material 24 . In addition, ϕ = tan −1 (Im[Z mea (ω)]/ Re[Z mea (ω)]) helps identify the suitable frequency for determining Z mea (ω → ∞) and Z mea (ω → 0) because ω R2C is unknown 15 . Finally, using the IS method, the value of zT was clearly estimated as (zT) IS = 0.811 (= (R ohm + R TE ) IS / (R ohm ) IS − 1 = 869.5 / 480.0 − 1). Figure 3 shows the results of the TH method for the same TM used in the IS method at 300 K for transient current I(t). R mea (t) was measured using a voltmeter (VM) and the remaining measurements were made using a data acquisition (DAQ) system. The inset in Fig. 3a shows the time dependence of R mea (t) for different values of current ranging from − 500 mA to + 500 mA. In addition, the current dependencies of R mea (t → ∞) at higher magnitudes of current due to the influence of Q J are clearly observable. I(t) using 1.447 mA at t ≥ 0 that is shown in Fig. 2a was selected from the essential condition of Q P (= 100 μW) > > Q J (= 1 μW) using (R ohm ) IS . Figure 2b shows that the value of R mea (t → ∞) = (R ohm + R TE ) TH,VM = 869.3 mΩ measured by the voltmeter was specifically determined owing to the large signal-to-noise ratio (SNR). R mea (t → ∞) was also measured at a different range of current, as shown in an inset in Fig. 3b, because R TE was replaced with R TE (1 + ηQ J /Q P ) in the higher current region in Eq. (1). It shows that R mea (t → ∞) is proportional to I (∝ Q J /Q P ) in the higher current region, as expected 16 . To avoid aliasing of the signals measured by the voltmeter, the sampling rate was set as 2.3 Hz; consequently, R mea (t → 0) = (R ohm ) TH,VM was ambiguous. Therefore, data acquisition had to be performed at a higher sampling rate to detect the variation of R mea (t → 0). The characteristic frequency ω R2C = 0.255 rad/s can be used as an approximate standard to derive the heat time constant τ exp of the system (~ 4 s = 1/ω R2C ). Figure 3c also shows R mea (t) obtained using a DAQ system at a sampling rate of 100 kHz, which would be sufficient to detect the transient response of R mea (t). Owing to the small SNR in the surroundings (or higher sampling rate), the data was averaged for a period ranging from 10 kHz to 1 Hz. However, detecting R mea (t = 0) from the results of the TH method is difficult despite the use of the DAQ. As the expected value of R mea (t = 0) was 480.0 mΩ from (R ohm ) IS , the first data for R mea (t = 0) was 471.27, 477.57, and 483.32 mΩ at 10 kHz, 1 kHz, and 100 Hz averaging, respectively. At a sampling rate of 100 kHz, the raw data of R mea (t → 0) was distributed from 23.94 to 554.62 mΩ during 10 μs. This result shows that detecting R ohm using the raw data of the TH method is difficult despite using the higher DAQ system. , respectively and phase angle ϕ at each AC (100 μA rms to 100 mA rms ) for the TM prepared. The upper axis shows normalized angular frequency ω/ω R2C . A lock-in amplifier and Quasi-AC method (implemented using a highprecision AC source and digital multimeter using real-time data acquisition for the low-frequency region) 12,28 were applied to measure the impedances at frequencies more than and less than 10 mHz, respectively. An inset shows its Nyquist plot of Z R2C (ω) and fitting plot by R2C approximation given in Eq. (2). www.nature.com/scientificreports/ Another method was developed to determine R ohm and R ohm + R TE by fitting the formula into the data obtained by the DAQ system. This proposal was based on the fact that R mea (t) is almost stabilized at 100 Hz averaging. Therefore, we used the value of R mea (t) measured by the DAQ using 100 Hz averaging in the TH method. This approach was adopted based on our hypothesis that the average frequency enables us to represent the transient response with τ exp . To estimate R mea (t → 0) = (R ohm ) TH,DAQ and R mea (t → ∞) = (R ohm + R TE ) TH,DAQ from Fig. 3c, two fitting equations for period Δt were applied from the equivalent circuit in Fig. 1d 16 . where τ R2C and τ RC are the estimated time constants using each equation, respectively. Figure 4 shows the calculation results from the period Δt. Figure 4a,b show R mea (t → 0) using the R2C (= (R ohm ) TH,DAQ,R2C ) and RC (= (R ohm ) TH,DAQ,RC ) approximations expressed in Eqs. (3) and (4), respectively. Near Δt ~ 0, both (R ohm ) TH,DAQ,R2C in Fig. 4a and (R ohm ) TH,DAQ,R2C in Fig. 4b were approximately 494 mΩ, and increased with increasing Δt owing to the excessive data available for R mea (t). Finally, (R ohm ) TH,DAQ,R2C and (R ohm ) TH,DAQ,RC were asymptotically close to 524.6 mΩ in Fig. 4a and 504.5 mΩ in Fig. 4b, respectively, which are approximately 1.09 and 1.05 times higher compared with (R ohm ) IS = 480.0 mΩ. The difference between (R ohm ) TH,DAQ and (R ohm ) IS obstructs the determination of (R ohm ) TH,DAQ when estimating zT. In other words, the value of zT is underestimated because (R ohm ) TH,DAQ > (R ohm ) IS . Therefore, estimating (R ohm ) TH,DAQ from R mea (t → 0) obtained using the TH method is unsuitable even if the DAQ was used for measurement. Furthermore, Fig. 4c,d show that (R ohm + R TE ) TH,DAQ converged at certain values, i.e., (R ohm + R TE ) TH,DAQ,R2C = (R ohm + R TE ) TH,DAQ,RC = 871.6 mΩ in Fig. 4c,d, which are consistent with (R ohm + R TE ) IS = 869.5 mΩ. The difference is acceptable because the difference is approximately 2.1 mΩ, which corresponds to the approximate 3 μV difference during the DC measurement. Moreover, the estimated time constants τ R2C and τ RC were 3.27 and 4.06 s, respectively, neither of which matched the estimated value. However, in the RC approximation using Eq. (4), τ RC = 4.06 s is not only close to τ exp = 1/ω R2C = 4 s as a representative heat transport time in this system, but also simpler to apply. Figure 4f quantitatively shows that the required period using the normalized time Δt/τ RC is more than 10 for each parameter. Furthermore, (zT) TH,DAQ was estimated to be 0.767 ± 0.014 (= (R ohm + R TE ) TH,DAQ,RC /(R ohm ) TH,DAQ,RC − 1 = 871.6/504.5 − 1), which is marginally smaller than that of the IS method because it is likely to overestimate (R ohm ) TH,DAQ compared with (R ohm ) IS .

Results
We also attempted to precisely measure R ohm using the pulse DC with an identical DAQ system because the R ohm term adds the contribution of the temperature difference generated for the DC. Figure 5 shows the results of measuring R mea (t → 0) as a function of t/T p , where T p is the pulse period. From Eqs. (1) and (4), the expression for R mea (t → 0) at Q P > > Q J can be derived as  3) and (4)  www.nature.com/scientificreports/ Figure 5. Normalized time dependence of (a) pulse current I(t/T p ), measured resistance R mea (t/T p ) at (b) t p = 100 ms and T p = 1000 ms, (c) t p = 10 ms and T p = 100 ms, (d) t p = 1 ms and T p = 10 ms for each pulse DC (1-10 mA ) by a DAQ system (100 kHz sampling rate), respectively. 10%, 40%, and 50% of the entire pulse period T p in (a) correspond to the pulse DC width t p , relaxation period to remove the temperature gradient (or difference) on the TEs of the TM, and offset period to determine the zero voltage for the next measurement, respectively. The inset in (d) shows how (R ohm ) pulse can be estimated from obtained data. www.nature.com/scientificreports/ Although the values of R mea (t/T p = 0) were approximately 480.0 mΩ (= (R ohm ) IS ) within the scattered data, it increased linearly at t p = 100 ms due to t p /τ RC = 2.46 × 10 -2 , as expected from Eq. (5). Moreover, if the value of zT is large, typically 1, the term zT × t p /τ RC in Eq. (5) should be less than 10 -3 to ensure precise R ohm measurement. The increase in R mea (t/T p ~ 0) was suppressed at shorter t p , and the variation of R mea (t) during t p was considered negligible owing to the considerably lower t p /τ RC , as shown in Fig. 5d for t p /τ RC = 2.46 × 10 -4 . Furthermore, current dependence was absent owing to a shorter t p , which possibly satisfies Q P > > Q J , during the t p /T p cycle. Finally, although the use of pulse DC was expected to be suitable for measuring (R ohm ) pulse , the measurement error within the scattered data near t/T p ~ 0 was also considered. Figure 6 shows the t p dependence of (R ohm ) pulse , and fell within the accepted error margin. It quantitatively shows that (R ohm ) pulse approached (R ohm ) IS = 480.0 mΩ at t p /τ RC < 10 -3 , as expected from Eq. (5). However, the measurement error of (R ohm ) pulse was prominent at all t p owing to the low SNR. Moreover, (R ohm ) pulse measured at a larger current, i.e., 10 mA, resulted in a smaller error compared with that measured at a smaller current; however, the error in the measurement of (R ohm ) pulse was still large. Based on this evaluation, pulse and transient Harman (PTH) method was proposed for determining (zT) PTH using (R ohm ) pulse and (R ohm + R TE ) TH,DAQ . (zT) PTH was estimated to be 0.804 ± 0.043 (= (R ohm + R TE ) TH,DAQ /(R ohm ) pulse − 1 = 871.6/483.2 − 1) using (R ohm ) pulse = 483.2 mΩ at I = 1 mA and t p = 5 ms (t p /τ RC ~ 10 -3 ), which is close to that of the IS method. Additionally, we attempted to measure (R ohm ) delta using the delta method, in which a current source is synchronized with an oscillating square wave at a frequency (a kind of pulse current) using a voltmeter 25 . Subsequently, the resistance can be measured after eliminating offset voltage and removing the influences of the thermoelectric-motive force in the circuit. The details are summarized in Supplementary Information. Furthermore, an important conclusion was drawn from the measurements that were used to determine (R ohm ) pulse using pulse DC in Figs. 5 and 6, (R ohm ) delta in Fig. S1, and R ohm + R TE using DC in Fig. 3c by the PTH method. The precise measurement of R ohm from R mea (t → 0) using only continuous DC is unsuitable for the TMs and TEs possessing large zT values despite the use of the DAQ system. This unsuitability can be attributed to the fact that large zT values affect the measurement noise (depending on the surroundings in the measurement system) even if the normalized pulse width t p /τ RC is less than 10 -3 . Although the transported Peltier heat Q P at I = 1 mA, shown Figs. 5 and 6 appeared small near Q P = 69.3 μW ( =|S|TI = 231 × 10 -6 × 300 × 1 × 10 -3 ) at 300 K 24 , the Peltier heat flux q P across the TE was not negligible, i.e., q P = 164 W/m 2 (= Q P /A = 69.3 × 10 -6 × (0.65 × 10 -3 ) −2 ) for A = 0.65 × 0.65 mm 2 owing to the generation of ΔT (~ q p L/κ).

Discussion
Based on the above-mentioned results and considerations, we proposed a combined measurement technique, namely, time domain impedance spectroscopy (TDIS), which attempts to measure R ohm = Z mea (ω → ∞) using a lock-in amplifier and AC at ω/ω R2C > 10 4 (or |ϕ|< 0.1°) and R ohm + R TE = R mea (t → ∞) with a voltmeter. Alternatively, it employs a DAQ system using DC at t/τ RC > 10 based on the TH method and partially derives the optimum current from the knowledge of the IS theory and model 10,11,16 . Consequently, (zT) TDIS was expressed as (zT) TDIS,VM = 0.811 ± 2.4 × 10 -4 (= (R ohm + R TE ) TH,VM /(R ohm ) IS − 1 = 869.3/480 − 1) using the voltmeter or (zT) TDIS,DAQ = 0.816 ± 3.6 www.nature.com/scientificreports/ × 10 -4 (= (R ohm + R TE ) TH,DAQ,RC /(R ohm ) IS − 1 = 871.6/480 − 1) using the DAQ (data of 100 Hz averaging). A marginal difference between (zT) TDIS,VM and (zT) TDIS,DAQ was derived from the measurement error of (R ohm + R TE ) TH,VM and (R ohm + R TE ) TH,DAQ,RC depending on SNR. The magnitude of the current I max (t > 0) required to suitably measure Z mea (ω > 10 4 ω R2C ) and R mea (t > 10τ RC ) should fulfil the condition I max < < R ohm /(n|S|T) to satisfy Q P > > Q J , where n is number of TEs in the TM (n = 1 is for the TE). In short, although its concept is relatively similar to that of the original Harman method 4 , the TDIS method accidentally revealed the method to measure Z mea (ω → ∞) and R mea (t → ∞) both quantitatively and qualitatively. Additionally, the temperature fluctuation ΔT f of the sample stage anchoring the TM during the measurements is an important factor. This ΔT f influences the measurement of the additional R TE (= n|SΔT|/I) from R ohm because the TDIS method is based on detecting ΔT derived from the Peltier effect (ΔT = R TE I/n|S|), given that ΔT > > ΔT f . Furthermore, all the experiments in this study were performed at a standard deviation of ΔT f ~ 0.3 mK in high vacuum (~ 10 -4 Pa) to maintain an adiabatic condition 26,27 .
We hypothesize that ΔT f /ΔT (~ 0.3 mK/174 mK) is a key factor that determines how accurately the value of zT can be detected by the TDIS 24 . Figure 7 and Table 1 show a summary of zT estimation using each technique. The (zT) IS estimated using the IS method, as shown in Fig. 2, was 0.811 ± 3.4 × 10 -5 . The result is reliable and (R ohm ) IS (= Re[Z mea (f = 513 Hz)]) was obtained by the lock-in amplifier within several seconds; however, the measurement of (R ohm + R TE ) IS (= Re[Z mea (f = 0.5 mHz)]) required several hours when employing the quasi-AC method in the lower frequency region (< 10 mHz) using the DC source and the voltmeter 12,28 . (zT) TH,DAQ estimated using the TH method, as shown in Figs. 3 and 4, was 0.767 ± 0.014. The estimated (zT) TH,DAQ differed from (zT) IS because determining (R ohm ) TH,DAQ (= R mea (t → 0)) was difficult despite applying the DAQ system. Finally, (zT) PTH estimated using the PTH method with pulse DC, as shown in Figs. 4, 5, and 6, was 0.804 ± 0.043, which was in moderately good agreement with (zT) IS . However, the measurement error to determine (R ohm ) pulse was prominent. In the TDIS method, to quickly determine R ohm and R ohm + R TE without error, (R ohm ) IS and (R ohm + R TE ) TH,VM or (R ohm + R TE ) TH,DAQ obtained from the lock-in amplifier and the voltmeter or the DAQ using the TH method with suitable AC and DC, respectively, were used. The zT values estimated by the voltmeter and the DAQ were (zT ) TDIS,VM = 0.811 ± 2.4 × 10 -4 and (zT) TDIS,DAQ = 0.816 ± 3.6 × 10 -4 , respectively. In addition, the error of (zT) TDIS,DAQ increased using raw data at a sampling rate of 100 kHz owing to the small SNR in Fig. 3d. We concluded that zT = ((R ohm + R TE )/R ohm − 1) can be determined by the TDIS method using R ohm = (R ohm ) IS = Z mea (ω → ∞) measured by the lock-in amplifier and R ohm + R TE = (R ohm + R TE ) TH,VM = R mea (t → ∞) measured by the voltmeter within www.nature.com/scientificreports/ several minutes. Precise measurements were obtained directly using a combination of AC and DC electrometric measurements without any heat measurements.
In this study, we reported the zT estimation using the TM owing to its higher resistance (~ 1 Ω) compared with a TE (~ 1 to 10 mΩ). In the future, the measurement technique of the TDIS method will be applied to determine zT using any TEs with a definite geometry, such as a rectangular solid TE. Furthermore, we plan to establish the accuracy of the TDIS method using the temperature dependence of the zT of BiTe, which decreases with temperature in the lower temperature region 12 .

Conclusions
In this study, we developed a new method to directly estimate zT. The proposed method is based on the theory and model of the IS method using frequency domain with a Π-shaped TM for a BiTe system. However, the IS method possesses several drawbacks, such as the long time required to determine zT. The reason behind this drawback is that the information about both Z mea (ω → ∞) and Z mea (ω → 0) using frequency domain with AC is required. In addition, we used the TH method using the time domain with DC, which is based on the time dependence of the resistance R mea (t), to measure the resistance R mea (t → ∞) and R mea (t = 0). However, the results showed that determining R mea (t → 0) using the TH method is difficult. Furthermore, we attempted to estimate R mea (t → 0) using the PTH method with pulse DC. However, we found that continuous DC was unsuitable for determining R mea (t → 0) because determining the resistance derived from the electronic and Peltier heat at R mea (t → 0) is difficult. To overcome these drawbacks, we proposed the TDIS method by combining the frequency and time domains to determine R ohm = Z mea (ω → ∞) using the lock-in amplifier with AC and R ohm + R TE = R mea (t → ∞) using the voltmeter with DC. Finally, zT was estimated as zT = R mea (t → ∞)/Z mea (ω → ∞) − 1 using optimum current I opt that satisfies the condition Q P (Peltier heat) > > Q J (Joule heat), given that I opt < <|S|T/R ohm . Furthermore, the estimated zT values of the TM using the IS and the TDIS methods were in perfect agreement, i.e., 0.811 at 300 K. Moreover, the TDIS method helped in qualitatively and quantitatively describing zT obtained from the IS method. We expect that this study will aid in developing more effective methods to determine zT precisely within several minutes for not only TMs but also any given TE.

Additional requirements
Optimum current I max (t > 0) < < R ohm /|S|T due to Q P > > Q J www.nature.com/scientificreports/ frequencies less than 10 mHz, implemented using a high-precision AC source and digital multimeter using real-time data acquisition for the low-frequency region 12,28 . The time dependence of the resistance R mea (t) was measured using a DC and pulse current source (6221, Keithley) for currents less than 100 mA, and a DC source (2400, Keithley), voltmeter (2182A, Keithley), and DAQ system (USB-6281, NI) for currents higher than 100 mA. All the instruments were connected through GPIB and USB cables and controlled appropriately by the LabVIEW (NI) program.

Data availability
Data is available upon reasonable request to the corresponding author.